Properties

Label 4-1148e2-1.1-c1e2-0-9
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 7-s − 4·9-s − 9·11-s + 2·14-s − 4·16-s + 8·18-s + 18·22-s + 4·23-s + 2·25-s − 2·28-s + 6·29-s + 8·32-s − 8·36-s − 12·37-s − 5·43-s − 18·44-s − 8·46-s − 6·49-s − 4·50-s − 15·53-s − 12·58-s + 4·63-s − 8·64-s − 9·67-s + 24·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.377·7-s − 4/3·9-s − 2.71·11-s + 0.534·14-s − 16-s + 1.88·18-s + 3.83·22-s + 0.834·23-s + 2/5·25-s − 0.377·28-s + 1.11·29-s + 1.41·32-s − 4/3·36-s − 1.97·37-s − 0.762·43-s − 2.71·44-s − 1.17·46-s − 6/7·49-s − 0.565·50-s − 2.06·53-s − 1.57·58-s + 0.503·63-s − 64-s − 1.09·67-s + 2.78·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 75 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.67335078497563088571091378899, −7.34541246957182503630415572812, −6.83498527129197968027715848030, −6.31267361837488764541791580866, −5.92040950196015795642402819102, −5.18741352189353919318546131999, −4.97798032650342694038900051396, −4.71033769091101380316359591352, −3.51887916480050099952252418650, −3.06192101833268841070385132312, −2.69224722252436758931654889310, −2.18035164361485404411932544620, −1.28053725312198844601612391556, 0, 0, 1.28053725312198844601612391556, 2.18035164361485404411932544620, 2.69224722252436758931654889310, 3.06192101833268841070385132312, 3.51887916480050099952252418650, 4.71033769091101380316359591352, 4.97798032650342694038900051396, 5.18741352189353919318546131999, 5.92040950196015795642402819102, 6.31267361837488764541791580866, 6.83498527129197968027715848030, 7.34541246957182503630415572812, 7.67335078497563088571091378899

Graph of the $Z$-function along the critical line